PLANETARY PERTURBATIONS. 213 Lagrange believed that these explosions had been frequent in our system, and that this was the true explanation of comets, judging from the greatness of their eccentricity and inclination, and the smallness of their masses. We have only to conceive that a planet may have burst into two very unequal fragments, the larger of which would proceed pretty nearly as before, while the smaller must describe a very long ellipse, much inclined to the ecliptic. Lagrange showed that the amount of impulsion necessary for this change is not great; and that it is less in proportion as the primitive planet is remote from the sun. This opinion is far from having been demonstrated; but it appears to me more satisfactory than any other that has been proposed on the subject of comets. Gradual perturbations. The important and difficult subject of perturbations is the principal object of celestial mechanics, for the perfecting of astronomical tables. They are of two classes; the one relating to motions of translation, the other of rotation. The latter are, as before, the most difficult: but the motions of rotation are less altered than the other class, within our own system; and they are less important to be known. In the study of motions of translation, the Perturbations planets must be treated as if they were condensed in their centres of gravity. of translation. The direct method, the only rational one, of calculating the differential equations of the motion of any one planet, under the influences of all the rest, is impracticable, from the unmanageable complication of the problem. It would make an inextricable analytical enigma. Geometers have therefore been obliged to analyze directly the motion of each planet round that which is its focus, taking for modification only one at a time. This is what constitutes in general the celebrated problem of three bodies, though this denomination was at first employed only for the theory of the moon. It is easy to see what circumvolutions are involved in this method, since the modifying body, being in its turn modified by others, compels a return to the study of the primitive body, to understand its perturbations. 'The determination of the motions of the whole of our system Problem of three bodies. must, by its very nature, be a single problem. It is the imperfection of our analysis which obliges us to divide it into detached problems, and to overload our formulas with multiplied modifications. The elementary problem of two bodies, one of these even being regarded as fixed,—is the only one that we are capable of bringing to a solution; the problem of the elliptical motion, represented by Kepler's laws; and here the calculations are extremely laborious. It is to this type that geometers have to refer the motions of the planets, by extremely complicated approximations, accumulating the perturbations separately produced by every body that can be supposed to exert any influence; and these perturbations prescribe the series required for the integration of the equations belonging to the case of the three bodies. Then follows the task of choosing the perturbations which have to enter into the estimate. The law of gravitation enables us to compare the secondary influences involved in each case, the masses of all within our own system being supposed to be known. It is a favourable circumstance to mathematical research that our system is constituted of bodies of very small mass in comparison with the sun (making the perturbations extremely small); moreover, very few, very far from each other, and very unequal in mass; the result of all which is that, in almost every case, the principal motion is modified by only one body. If the contrary had been the case, the perturbations must have been very great, and extremely varied, since a great number of bodies must have powerfully acted in each disturbance. Celestial Mechanics must then, we should think, have presented an inextricable complication, being incapable of reduction to the problem of three bodies. The study of modified motions divides itself into three parts, answering, as in a former case, to the planets, satellites and comets. Rigorously speaking, we ought to make a fourth case of the sun, which cannot here be regarded as motionless, because the planets react upon it. In fact, we cannot allow ourselves to consider any point within the system as motionless, except the centre of gravity of the system itself, which is the true focus of planetary motion, and round which the sun Centre of the THE THREE PROBLEMS. 215 itself must oscillate, in directions which vary according to the positions of the planets. This point is always between the centre and the surface of the sun. But we cannot approach nearer to the fact than this: we shall probably never be able to indicate this centre precisely; and it is enough for practical purposes, and necessary to them, to consider the sun as fixed, except as to its rotary motion. The same conclusion must be come to with regard to the planets and their satellites, even in the case of the earth and moon, where the variations of the primary body are greatest. The centre of gravity falling within the mass of the primary body, its variations from that centre may be neglected as having no appreciable influence on the motion of translation; and thus, celestial mechanics presents, in this branch, no other problems than those treated, under another point of view, by celestial geometry. Problem of the Planets. The simplest problem is here, as before, that of the planets, and for the same reasons, -the smallness of their eccentricities, and of the inclinations of their orbits. There is also a considerable uniformity of perturbations, since each planet remaining in the same regions of the sky, continues in the same mechanical relations, though their intensity varies within certain limits. The least privileged of these bodies in these matters is unhappily our own planet, on account of the heavy satellite which escorts it so closely, and to which its chief perturbations are due; though this does not save it from being sensibly troubled by others, at the period of opposition, and especially by such a mass as that of Jupiter. No other planet with satellites, not even Jupiter, is in so unfavourable a case; for Jupiter's motion could not be very much deranged by the action of his satellites, however near in position, since the mass of the largest is less than a ten-thousandth part of his, while the mass of our moon is a sixty-eighth part of that of the earth. Jupiter's circulation is sensibly affected by Saturn alone. The simplest case of all seems to be that of Uranus, from its being the last planet, and very remote from the next; and its six satellites do not appear to trouble its motion. The problem of the satellites is necessarily Problem of more complicated than that of the planets, the Satellites. on account of the instability of the focus of the principal motion, as in celestial geometry. Besides their own perturbations, the satellites have reflected upon them all those to which their planet is liable. The founders of Celestial Mechanics were long perplexed, for instance, by the perpetual acceleration of the mean motion of the moon; it was considered inexplicable, till Laplace discovered its cause in the slight variation to which the eccentricity of the earth's orbit is subject. In regard to the direct perturbations of the satellites, there is an essential distinction between the case of one, and that of several satellites. In the first, the single case of our moon,—the disturbing body is the sun, on account of its unequal action on the planet and the satellite. If the difficulties arising out of this position are greater than in the case of any other satellite, it is partly because the case more immediately concerns us, and because our opportunities of observation disclose more fully the imperfection of our means. For, in the mathematical point of view, there must be more complexity in the case of several satellites; all that is true in regard to one being true in regard to each one, with the addition of the mutual action of the members of the group. Their perturbations are reduced by the preponderating size of their planet; but from there being so many of them, of such nearly equal sizes and direction, and all so close together, the difficulty of calculating their motions is so great that the only theory as yet established is that of the satellites of Jupiter. For the motions of three of them, Laplace found means completely to account. Those of Saturn and Uranus are known only geometrically, we having not even an approximate estimate of their masses. It is to be remembered, however, that we do not need so perfect a knowledge of them as of the moon; and that a much less exact theory will suffice for them than for the moon, whose slightest irregularity is very evident to us. Problem of the Comets. The comets intervene to increase our difficulties about the satellites. From the extreme prolongation of their orbits, and their inclination in all directions, comets are in a state of ever variable mechanical relations, from the number of bodies that they approach in their course; whilst the planets, and PROBLEM OF THE COMETS. 217 even the satellites, have always the same relations, the variation being only in the intensity. The perturbation which, in every other case, bears a very small proportion to the gravitation, may, in the case of comets, exceed it: so that it is conceivable that a comet might be diverted from its orbit, and become a satellite, when it passes near so considerable a body as Jupiter, Saturn, or even Uranus. Besides the eccentricities of comets, there are other circumstances, such as their small weight, and their possible loss of weight by parting with some of their atmosphere to the bodies they approach, which tend to perplex the study of their perturbations. These are the incidents which make it so difficult to foresee exactly the return of these little bodies. When we have studied them so long and so laboriously as to have, to the best of our belief, mastered their case, we find that their periods are entirely changed through one omitted circumstance. A memorable example of this was the comet of 1770, calculated by Lexell. This comet had then a revolution of less than six years: but it has never appeared since, having been entirely deranged by passing too near Jupiter. The imperfection of our knowledge about these small bodies is from the same cause that renders them of very little consequence to us. From their vast distances, their action upon any one body of the system is little more than momentary; and their lightness prevents even the satellites from being affected by their passage. The passage of the comet of 1770 among the satellites of Jupiter proved this, in a striking manner. Their tables, constructed beforehand, without any idea of such an incident, perfectly agreed with direct observations; a proof that the intrusion of the comet did not sensibly affect their motions. There is, therefore, no more occasion for the puerile fears of our day than for the religious terrors of former times, in regard to the passage of comets. Their collision with the earth is all but impossible; and they could not otherwise be felt at all. Their mere approach, however near, could have no other effect than to raise somewhat the corresponding tide. If a comet could pass two or three times nearer to us than the moon (which no known comet could do) its very small mass could produce no other effect than an imperceptible |